Autoregressive models with distributed lags (ADL) It often happens than including the lagged dependent variable in the model results in model which is better fitted and needs less parameters. It can be explained by the inertia of many economic processes Model in which we have lagged explanatory variables is called autoregressive model General class of such a models are ADL models (Autoregressive Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 1
Distributed Lags) of the form: y t = α 1 y t 1 +... + α p y }{{ t p } AR + µ + x t β 0 + x t 1 β 1 +... + x t s β }{{} s DL + ε t Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 2
Impact and long tun multipliers in ADL Impact multiplier is the same os in DL models and is equal to E ( y t ) x t = β 0 Log run multiplier can be calculated in a following way. We change x t, x t 1,... by x. The expected change of y t is equal to: E (y t + y t ) = α 1 E (y t 1 + y t 1 ) +... + α p E (y t p + y t p ) +µ + (x t + x) β 0 + (x t 1 + x) β 1 +... + (x t s + x) β s Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 3
and then y t is equal to E ( y t ) = α 1 E ( y t 1 ) +... + α p E ( y t p ) + x (β 0 + β 1 +... + β s ) But in long run the influence of permanent change of x on y stabilizes: E ( y) = E ( y t ) = E ( y t 1 ) =... = E ( y t p ) So the long run multiplier β is equal to: E ( y) x = β = β 0 +β 1 +... + β s 1 α 1... α p Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 4
Example 1. Unemployment and money supply (cont.). We introduce the lagged variable unemployment as explanatory variable and eliminate insignificant lags of explanatory variables. Source SS df MS Number of obs = 42 -------------+------------------------------ F( 2, 39) = 246.00 Model 452.505732 2 226.252866 Prob > F = 0.0000 Residual 35.8693916 39.91972799 R-squared = 0.9266 -------------+------------------------------ Adj R-squared = 0.9228 Total 488.375123 41 11.9115884 Root MSE =.95902 ------------------------------------------------------------------------------ bael Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- bael L1.8508812.0571166 14.90 0.000.735352.9664103 m3p -.1570696.0519469-3.02 0.004 -.2621422 -.051997 _cons 3.015409 1.04154 2.90 0.006.9086948 5.122124 ------------------------------------------------------------------------------ Obtained model is better fitted but has less parameters Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 5
Impact multiplier of the money supply is equal to: β 0 =.1570696 Long run multiplier of the money supply is equal to: β =.1570696 1.8508812 = 1. 053 3 Long run multiplier is approximately the same as in the DL model! Breusch-Godfrey test: Breusch-Godfrey LM test for autocorrelation --------------------------------------------------------------------------- lags(p) chi2 df Prob > chi2 -------------+------------------------------------------------------------- 1 0.309 1 0.5782 --------------------------------------------------------------------------- H0: no serial correlation No autocorrelation - dynamics of unemployment seems to be well described! Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 6
Long run equilibrium Long run equilibrium (long run solution) it the state in which the expected value of the depend variables is constant if the exogenous variables are constant Long run equilibrium is interesting from the point of view of the theory because the majority of macroeconomic theories concern the relations between the economic variables in equilibrium The equilibrium solution we find using its definition, by assuming that the expected value of the dependent variable and independent variables are Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 7
constant over time y = E (y t ) = E (y t 1 ) =... = E (y t p ) x = x t = x t 1 =... = x t s Substituting into the definition of ADL we get: (1 α 1... α p ) y = µ + x β 0 + x β 1 +... + x β s For model ADL it implies that long run equilibrium is given by: y = µ + x β where µ = µ 1 α 1... α p a β = β 0 +β 1 +...+β s 1 α 1... α p. Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 8
Notice that β is exactly equal to long run multiplier Example 2. Unemployment and money supply (cont.). The long run equilibrium in this model is given by: y = 3.015409 1.8508812 +.1570696 1.8508812 x = 20. 222 1. 053 3x For money supply expansion equal to zero, the unemployment would be equal to 20.222% Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 9
Causality testing Haw the cause and the result can be defined? Features of causality: cause always is precede the result if the cause take place we can predict that the result will take place as well Granger causality (very special definition of causality cannot be taken as definition of causality in philosophical sense) Definition 3. Variable x is Granger cause of y if current values of y can be better forecasted with the use of past x s than without them. Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 10
In model y t = a (t) + k α i y t i + i=1 k β i x t i + ε t where a (t) is deterministic part of the model (e.g. a (t) = γ 0 + γ 1 t), if β 1 = β 2 =... = β k = 0, then x is not a Granger cause of y Granger causality test: We test the null hypothesis that x is not a Granger cause of y by testing joint hypothesis that all β i related to lagged x are equal to zero H 0 : β 1 = β 2 =... = β k = 0 Example 4. Unemployment and money supply (cont.). In order to test Granger causality we should build the model with only lagged values of the variable the causality of which we are testing (contemporaneous x s should not be included) i=1 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 11
Source SS df MS Number of obs = 42 -------------+------------------------------ F( 2, 39) = 200.84 Model 445.153629 2 222.576814 Prob > F = 0.0000 Residual 43.2214946 39 1.10824345 R-squared = 0.9115 -------------+------------------------------ Adj R-squared = 0.9070 Total 488.375123 41 11.9115884 Root MSE = 1.0527 ------------------------------------------------------------------------------ bael Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- bael L1 1.007958.0673853 14.96 0.000.8716585 1.144258 m3p L1.0603557.061816 0.98 0.335 -.0646789.1853903 _cons -.3409437 1.249234-0.27 0.786-2.867759 2.185871 ------------------------------------------------------------------------------ In this simple model we any need to test the significance of lagged m3p According to test result, changes of money supply are not Granger cause of changes in unemployment! Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 12
Knowledge of the changes of money supply in the past cannot be used for improving the forecasts of the future unemployment. Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 13
Consistency of OLS in estimation of ADL Variables and ADL model should be predetermined (exogenous or lagged endogenous) If there is an autocorrelation in the model with lagged depend variables, then we simultaneity will occur In this case the b OLS will not be consistent Because of this problem we should always try to eliminate the autocorrelation from the model wit autoregressive part For testing autocorrelation we can use Breusch-Godfrey test Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 14
In model with autoregressive part the DW test cannot be used for testing autocorrelation as it is consistent Autocorrelation can normally be eliminated by adding lags of dependent variable to the model Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 15
Lag operator We define L as As x t 1 = Lx t LLx t = Lx t 1 = x t 2 = L 2 x t, the s power of L can be defined as such operator that L s x t = x t s As [ 1 + al + (al) 2 +... (al) n] (1 al) = 1 (al) n+1 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 16
then 1 + al + (al) 2 +... (al) n = (1 al) 1 [ 1 (al) n+1] If a < 1 then (1 al) 1 = (al) i i=1 Conclusion: 1 al can be inverted, if a < 1 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 17
Polynomials of lag operator Polynomial of lag operator: A (L) = 1 a 1 L a 2 L 2... a s L s Polynomial A (x) = 1 a 1 x a 2 x 2... a s x s can be factorized into A (L) = (1 λ 1 L)... (1 λ s L) where µ i = λ 1 i are root of this polynomial (possibly complex numbers) Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 18
1 λ i L can be inverted if λ i < 1. A (L) can only be inverted if λ i < 1 (or µ i > 1) for i = 1,..., s A (L) can be inverted only if all its roots lie outside the unit circle The A (L) 1 can the be written as A (L) 1 = (1 λ 1 L) 1... (1 λ s L) 1 [ ] [ ] = (λ 1 L) i... (λ s L) i = i=1 ψ i L i i=0 i=1 where ψ 0, ψ 1, ψ 2,... are parameters converging to zero. Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 19
Example 5. ADL model written with the use of polynomials of lag operator: y t α 1 y t 1... α p y t p = µ + x t β 0 + x t 1 β 1 +... + x t s β s + ε t where A (L) y t = µ + B (L) x t + ε t A (L) = 1 α 1 L... α p L p B (L) = β 0 β 1 L... β s L s If all the root of polynomial A (L) lie outside the unit circle we can write y t = µ + A (L) 1 B (L) x t + A (L) 1 ε t = µ + ψ i x t i + φ i ε t i i=0 i=0 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 20
ADL model is equivalent to DL model with infinite number of lags and autocorrelated (moving average) error term! Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 21
Difference operator Definition so 1 L. x t = x t x t 1 = (1 L) x t Differences of order p we denote as p = (1 L) p For instance 2 x t = (1 L) 2 x t = ( 1 2L + L 2) x t = x t 2x t 1 + x t 2 = x t x t 1 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 22
Seasonal differences: For quarterly data: s = 1 L s 4 x t = x t x t 4 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 23
Seasonality Seasonality take place if we observe cyclical changes in data related to seasons of the year For example quarterly data often shows quarterly seasonality and monthly data monthly seasonality. Seasonality in the model can be taken into account into model in two ways: we can include in the model additional dummy variables related to each quarter (month) we can use seasonal differencing: for example in the case of quarterly data 4 y t = y t y t 4 Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 24
Example 6. GDP in Poland in years 1996-2004. Seasonal changes in this case are related to investment cycle - the largest investments are registered in forth quarter. Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 25
rpkb 1400 1600 1800 2000 2200 1995q1 1997q3 2000q1 2002q3 2005q1 date P KB in real terms - CP I deflator Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 26
S4.rpkb 0 50 100 150 1995q1 1997q3 2000q1 2002q3 2005q1 date P KB in real terms, seasonally differenced Macroeconometrics, WNE UW, Copyright c 2005 by Jerzy Mycielski 27